Supplementary Materials for

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1 Supplementary Materials for Evidence for the chiral anomaly in the Dirac semimetal Na 3 Bi Jun Xiong, Satya K. Kushwaha, Tian Liang, Jason W. Krizan, Max Hirschberger, Wudi Wang, R. J. Cava, N. P. Ong* *Corresponding author. npo@princeton.edu This PDF file includes: Materials and Methods Supplementary Text Figs. S1 to S6 Published 3 September 1 on Science Express DOI:.116/science.aac689

2 In the main text, we report evidence for the chiral anomaly in the Dirac semimetal Na 3 Bi. In Sec. S1, we review the derivation of the Dirac Hamiltonian and the chirality of Weyl nodes in Na 3 Bi. Section S reviews the resistivity and conductivity tensors in a tilted B for a conventional one-band anisotropic metal. We report supplemental information from Sample J (Sec. S3), and discuss similar results from a second sample J1 (Sec. S). In Sec. S, we discuss the Weyl-node shift arising from the g-factor in a field. In Sec. S6, we discuss why localization is not the cause of the observed negative magnetoresistance. Throughout, we adopt the notation, x-y axes and contact labels defined in the inset in Fig. 1C of the main text (and reproduced as an inset in Fig. S1A). S1. CHIRALITY OF WEYL NODES IN Na 3Bi We sketch the essential steps in deriving the Dirac Hamiltonian and calculating the chirality of the Weyl nodes. According to Wang et al. (Ref. 1 of main text) the electronic states closest to the Fermi energy are dominated by the Na-3s (in the conduction band) and Bi-6p x and Bi-6p y states (valence band). The large spin-orbit coupling (SOC) in Bi leads to band inversion by. ev, which produces band crossings of the s and p states along Γ-A or (1) direction. Because the states belong to different irreducible representations, they cannot hybridize. This results in protected Dirac nodes at K ± = (,, ±k D ). The protected crossing involves only the states S + 1, ± 1 and P 3, ± 3. (Here the superscripts indicate the parity of the linear combinations of orbitals centered at the two ions or cations in the unit cell. The subscripts are the total angular momentum J of the orbital.) In the basis ( S, 1, P, 3, S, 1, P, 3 )T where S, ± 1 is shorthand for S+ 1, ± 1, etc, the Hamiltonian for states very close to the node at K + reduces to H = v k z k + k k z k z k k + k z (S1) with v the Dirac velocity ( = 1). The wave vector k is measured relative to the node at K + (with k ± = k x ± ik y ). For simplicity, we have retained only the leading terms and assumed isotropic dispersion. The Hamiltonian in Eq. S1 describes massless Dirac states with energy E = v k and a -fold degeneracy at E =. Clearly, H breaks up into two blocks which we identify with the Weyl nodes, both centred at K +. Using the (orbital space) Pauli matrices (τ 1, τ, τ 3 ), we write the upper block as H 1 = k ṽ 1 τ = v(k x τ 1 k y τ +k z τ 3 ), where the velocity matrix is given by ṽ 1 = v (S) The chirality of Weyl nodes is defined by (Ref., main text) χ = det[ṽ]/v. (S3) Hence Eq. S gives χ 1 = 1. For the lower block, we have H = v( k x τ 1 k y τ + k z τ 3 ), from which we get χ = +1. Thus, in zero B, the Dirac node may be viewed as the superposition of two Weyl nodes with opposite χ. The two populations (characterized by their charge χ) do not mix and are separately conserved. However, application of the parallel fields B and E breaks the chiral symmetry, causing chiral charge to be pumped between the two Weyl nodes. This is the chiral anomaly (Refs., 7 and 11, main text). S. RESISTIVITY AND CONDUCTIVITY TENSORS IN TILTED FIELD We address the question of what may be expected for the MR in the standard semiclassical, one-band model in a tilted field. For a single anisotropic band, the Boltzmann equation is ee v f k ϵ k + ev B g k k = g k τ, (S) with e the charge, E the electric field, v = ϵ k / k the band velocity, g k the leading correction to the equilibrium distribution function fk, and τ the relaxation time. We take the magnetic field to be in the x-zplane, i.e.

3 y 3 we can solve Eq. S to get the resistivity tensor (n is the carrier density) ˆρ = ρ 1 B 3 /ne B 3 /ne ρ B 1 /ne. (S7) B 1 /ne ρ x Inverting ˆρ, we obtain the conductivity tensor ˆσ = 1 σ 1(1 + µ 1 µ 3 B1) σ 1 µ B 3 σ µ 1 µ 3 B 1 B 3 σ 1 µ B 3 σ σ µ 3 B 1 (S8) σ µ 1 µ 3 B 1 B 3 σ µ 3 B 1 σ 3 (1 + µ 1 µ B3) where the zero-field conductivities and resistivities are defined by σ i = 1/ρ i = neµ i in terms of the anisotropic mobilities µ i = eτ/m i. The determinant equals (1 + µ µ 3 B1 + µ 1 µ B3). Inspection of the entries shows that, despite the anisotropy and the field tilt, the longitudinal resistivity ρ xx is strictly independent of B (the Hall E-field perfectly balances the Lorentz force). In the limit B 3 (or θ in Fig. C,D of main text), the longitudinal conductivity σ xx acquires a mild field dependence that vanishes if µ 1 = µ = µ 3. It never diverges as 1/ sin θ. Equations S7 and S8 also apply to the geometry with B in the x-y plane (Fig. 3 and Fig. A,B of main text) if we exchange the axes y z. S3. ADDITIONAL RESULTS IN SAMPLE J FIG. S1: The unsymmetrized magnetoresistance data in Sample J. Panel (A): The unsymmetrized MR data at. K derived from resistance R 1,3 (current I applied to the contacts (1,) and voltage measured between contacts (,3)). The inset shows the contact labels and the x and y axes fixed to the crystal. Panel (B): The unsymmetrized MR data based on resistance R 3,6. The raw data has a small antisymmetric part that we attribute to a small Hall signal inadvertently caused by misalignment between the field tilt plane and the crystal s a-b face. Despite this small Hall signal, the negative MR is prominent in both cases when ϕ (or ϕ ) is close to zero. B = (B 1,, B 3 ). Using the ansatz with u the drift velocity g k = u k f k ϵ k, and the effective mass matrix ˆm = m 1 m m 3 (S) (S6) In semimetals with small carrier density n and high mobility µ >, cm /Vs, the magnitude of the Hall resistivity ρ yx can greatly exceed the longitudinal resistivity ρ xx (in the standard Hall geometry with current I ˆx and B z). In most of the results in the main text and supplement, B is kept in the x-y plane. Theoretically, the Hall effect should be zero. In experiments, however, the Hall signal is often picked up as a background signal that can become comparable with the resistive voltage above Tesla. This arises because, for instance, the rotation plane is slighly misaligned with the crystal s a-b plane, unintentionally introducing a z component to B. For serious misalignments, this can lead to a very strong distortion of the raw MR curves, causing them to tilt dramatically towards, say, the +B axis. We believe this is a major source of negative resistance (E x < for I ˆx) sometimes encountered in semimetal magnetoresistance experiments performed in large B. Such distortions can mimic a negative MR when the curves are symmetrized in field. Raw curves To demonstrate that this distortion is a relatively weak effect in our experiment, we display the raw curves (pre-symmetrization). Figure S1A displays the unsymmetrized raw MR curves measured in Sample J for I applied to contacts (1,) (the measured resistance is R 1,3 ). Figure S1B shows the corresponding rotated curves when I is switched to the contacts (3,) (measured resistance R 3,6 ). In Panel A (B), ϕ (ϕ ) denotes the

4 3 yx (m cm) T=. K B (T) Sample J FIG. S: The Hall effect data of J at different temperatures. The Hall curves at various temperatures reveal a clear signature of thermally excited holes at high temperatures. Above 7 K, the holes start to contribute to the Hall effect and they finally dominate the Hall signal above K. The conductivity enhancement in MR also disappears around the same temperature range. angle between B and ˆx (ŷ). The asymmetry of the MR curves in both panels is caused by a Hall pick up signal. However, it is too small to be the origin of the large, negative MR observed when B is aligned with I. Hall curves Figure S displays the temperature dependence of the Hall curves in Sample J measured with I applied to the contacts (1,) and B parallel to the c- axis. As shown in figure, electrons dominate in the sample at low temperatures. But the hole contribution grows rapidly with an increasing temperature. We observed a clear contribution from the holes at a small B above 7 K. Then the holes completely dominate the Hall effect at temperatures above K. This hole excitation behavior is consistent with our picture (Fig. 1B of main text), in which the Fermi energy ϵ F 3 mv is near the Dirac node. When T is raised above K, the steep increase in both electrons and holes to higher Landau levels (which are not chiral) dominates the conduction, rendering the axial current difficult to resolve. Background subtraction for SdH curves We explain our procedure for subtracting the background from the observed MR curves to isolate the SdH oscillations. Restricting attention to the field range < B < H k, we first fit the observed curve to the quadratic polynomial ρ bg (B) = c + c 1 B + c B (H k is the kink field shown in Fig. 3D in the main text). After subtracting the polynomial, the SdH oscillations become well-resolved in the trace ρ xx (B) ρ xx (B) ρ bg (B). (S9) Some of these curves are plotted in Fig. S3 for θ = 9 (Panel A) and 6 (Panel B). In each panel, various polynomial choices have been used. Slight differences in the coefficients c 1 and c can cause the SdH amplitudes to vary by up to a factor of. To label each curve, we quote (on the right) the percentage values representing the amplitude, measured from the dashed line, of the largest observed peak (at B n ). The amplitude is expressed as a fraction of the total resistivity swing ρ xx (B n ) ρ xx (). As shown, the amplitudes vary significantly between curves. However, the values of extremareciprocal fields 1/B n, plotted in Fig. B in the main text, are barely affected (shifts are less than %). This is the main source of uncertainty in the Landau index plot. S. MAGNETORESISTANCE IN SAMPLE J1 The transport results on a second crystal, Sample J1, show behavior similar to that of J described in the main text. Sample J1 was crystallized within the same boule as J. The inset in Fig. S shows the dependence of J1 s resistivity ρ on temperature T. As in J, ρ rises to a peak below K as the hole contributions are frozen out. The value of ρ at K is about smaller than in J. The longitudinal magnetoresistance inferred from measurements of R 1,3 with B ˆx are shown in the main panel. For weak B (< T), we observe a large negative MR peak centered at B= that is suppressed when T is raised from 3. to K. This pattern identified with a large axial current in the lowest Landau level (LL) that becomes overwhelmed by contributions from higher LLs at elevated T is similar to the behavior observed in J. For the MR curves at very large B (> T), however, the distortion from an unintended z component B z become apparent. (Because the measurements on J1 preceded those performed on J, we were unaware of the critical alignment conditions needed to isolate the anomalous current, so the z component of B is quite significant, even as the field is rotated in the nominal x-y plane. An angular estimate of the misalignment is given below.) The main panel in Fig. S plots the magnetoresistance (MR) derived from the resistance R 1,3 at fixed T from 3. to K measured with B ˆx. In Fig. S6A, we show curves of the conductivity σ xx (in J1) measured with B lying in the x-y plane (at an angle ϕ to ˆx). The curves are derived from measurements of R 1,3. As in J of the main text, the conductivity displays a strong B enhancement when ϕ is close to consistent with Eq.. In Panel B, we show curves of the resistivity ρ xx measured up to 3 T. Again, similar to J, the MR is large and positive when B is transverse to I (curves at 8 ), but becomes negative as ϕ. From these curves, we estimate the angular misalignment in J1 to be ( ± 3). A second consequence of the misalignment appears above 3 T. As in J, we observe a knee feature at the kink field H k 3-3 T, above which ρ xx rises steeply. In J, H k strongly disperses with ϕ. When ϕ < it increases

5 A xx (m cm) B xx (m cm) =9 o /B(T -1 ) %.% 1.%.8 =6 o % %.% /B(T -1 ) FIG. S3: Comparison of the SdH curves measured at θ = 9 (Panel A) and 6 (Panel B) in Sample J after subtraction of a polynomial background ρ bg to isolate the oscillations (see Eq. S9). Depending on the particular polynomial adopted, the resulting SdH amplitudes (measured relative to the dashed line) can vary by up to. We label each curve by the amplitude of the peak at the highest field, expressed as a percentage of the total resistivity swing between B = and H k ; see text. However, unlike the amplitudes, the values of 1/B n at each extrema (numbers attached to arrows) are barely affected, shifting by less than %. above 3 T (see Fig. D of main text). Hence the ϕ = curve is unaffected. By contrast, H k (ϕ) has a milder variation vs. ϕ in Fig. S6B. Specifically, in the curve, an upturn is apparent above 3 T. We interpret this to imply that a sizeable B z component exists for the curve measured at ϕ =. The main features of the MR curves reported in the main text are also observed in J1. However, the larger (m cm) 6 Sample J T (K) FIG. S: Temperature dependence of the resistivity ρ in Sample J1 in zero B. The non-metallic profile, closely similar to that in J, is consistent with the freezing out of hole excitations as T K. (m cm) xx 8 6 Sample J1 R 1,3 T= 3. K B (T) FIG. S: Large negative, longitudinal MR at 3. K observed by measuring R 1,3 with B ˆx. The negative MR at low fields implies strong increase in the conductance when B E. As T is raised to K, the negative MR anomaly is progressively suppressed as the conductance of states in upper Landau levels become dominant. The very small increasing background discernible above 7 T reflects a small B z caused by misalignment. misalignment of the crystal plane in J1 engenders an upturn in ρ xx above 3 T that is not present in J. Comparison of the results from J1 and J in the large-b limit provides a useful estimate of the angular width of the collimated beam associated with the axial current. 1

6 A xx cm -1 B xx (m cm) 8 6 B (T) Sample J1, T = 3. K Sample J1, T = 3. K R 1, B(T) = - o = 6 o FIG. S6: Panel (A): Variation of the conductivity curves σ xx(ϕ) vs. B at selected tilt angles ϕ in Sample J1 at T = 3. K. σ xx (ϕ) is derived from measurements of R 1,3. As ϕ is rotated to close to, σ xx increases strongly as B consistent with Eq. of the main text. We identify this with the axial current. Panel B shows the curves of ρ xx vs. B measured in J1 at 3. K at selected tilt angles ϕ in fields up to 3 T. The profiles are similar to those shown in Fig. D of the main text for Sample J. A kink feature appears at H k in the rangle 3-3 T, above which ρ xx increases steeply. Here, the ϕ = curve (for J1) is also affected by H k (unlike in J), possibly because of the larger misalignment in J1. S. SHIFT OF WEYL NODE WITH FIELD In a field B ẑ, we have k Π = k ea, with the vector potential A = xbŷ. From Eq. S1, the Weyl Hamiltonian H 1 is then (reinstating ) [ ] [ ] Πz Π H 1 = v + gs µ Π Π B B, (S) z g p where Π ± = Π x ± iπ y and µ B is the Bohr magneton. Expressing the g-factors g s and g p (which are distinct) of the starting atomic orbitals S, ± and P, ± as g s = g m + δg and g p = g m δg, we have the Hamiltonian H + = vπ τ g m µ B B1 δgµ B Bτ 3, (S11) where τ i are the Pauli matrices in orbital space. By transforming the Π ± to raising and lowering operators a and a, with l B the magnetic length, a a Π + =, Π = (S1) l B we diagonalize H + to obtain the eigenenergy of the n th Landau level n E n (k z ) = g m µ B B ± v + ( vk z δgµ B B). (S13) Setting the second term under the square root to zero, we obtain k = δgµ B B/ v. This is used in the main text to estimate the shift. If g m 1, the term g m µ B B in Eq. S13 can lead to strong deviation of the index plot from a straight line in Fig.. S6. CAN LOCALIZATION ACCOUNT FOR THE OBSERVED NEGATIVE MR? The phenomenon of negative MR is a long-standing problem in metals. Although Anderson localization is arguably the most well-established theory for negative MR in D disordered metals, we believe it is irrelevant to the negative LMR in Na 3 Bi. We discuss the reasons here. First, we briefly review the salient facts about localization (for reviews, see [1, ]). We consider electrons diffusing in a disordered metal at very low T. The amplitude A for transmission between points P and P is the sum of contributions over all Feynman paths linking P to P. The subset of paths that self-intersect to form closed loops make singular contributions to A which decrease the conductivity σ from its Drude value σ in zero magnetic field. A modest field B suppresses the quantum corrections, restoring σ. Hence one observes a negative MR. [The diffusive nature of electrons moving in a sea of impurities dictates that the size L of a closed loop is generally l, the electron mean-free-path. In zero B, the paths traversing a closed loop (clockwise vs. anticlockwise) are exact time-reversed partners. Hence wave packets following the two paths must interfere constructively when they return to the node regardless of L. Consequently, the amplitude for observing the electron at the node is enhanced, leading to Anderson localization. Because raising T increases inelastic scattering by phonons (which destroys the quantum coherence), L shrinks rapidly with increased T. As T decreases, the quantum corrections lead to a resistance R that increases as log T in D. For 3D metals, R varies as T n (n = 1 1 ). In finite B, the constructive interference is suppressed once the magnetic flux in L exceeds 1 a flux quantum ϕ. As noted, this leads to a negative MR, with R log B in D. In 3D, R B 1. In D l B l B

7 6 metals, the characteristic field decreases from G at K to G at mk, as L increases. Experimentally, D localization has been extensively investigated [1, ]. In 3D metals, however, it has been difficult to observe in MR experiments [1]. The corrections are weak presumably because Feyman paths in 3D rarely self-intersect.] The reasons why we believe localization is not relevant here are as follows. First, we consider the ρ vs. T profile in B = (Fig. 1C, main text). As T decreases from 3 K, ρ rises by a factor of 1, but saturates to a constant below K. The large variation ρ between 3 and K is matched by a large change in R H, which implies that ρ is actually driven by copious excitation of holes in a zero-gap semimetal rather than by Anderson localization (which should not be observable above a few K in 3D). The abrupt saturation below K is also incompatible with localization which predicts that ρ should increase with ever steeper slope as T (as L expands). The pronounced angular anisotropy of the axial current is incompatible with localization in a 3D metal. As shown in Figs. 3 and (main text), the large negative MR is observed only when B is aligned to within 3- of I. This highly anisotropic MR variation cannot be reconciled with 3D localization (the quantum interference effects underlying 3D localization imply that the negative MR should be isotropic in B). Also, the negative MR persists to 9 K, far higher than the K typical in 3D localization experiments. The overall change of ρ xx in the negative MR profile (Fig. S and Fig. 1D of main text) is very large ρ xx decreases by a factor > at K compared with a few % in typical D localization experiments [1, ] (in 3D, the change is even smaller). Quantitatively, we note that the field scale B w needed to suppress the peak in ρ xx is 3- T, compared with < Oe typical in localization experiments. If localization were indeed the cause of the negative MR, we would require typical loop size L = ϕ /B w nm at K, but such an L would be far too small to produce the observed change in ρ xx. Moreover, this L implies a very short l which is inconsistent with the measured mobility (µ = 6 cm /Vs at K). * * *